Step |
Hyp |
Ref |
Expression |
1 |
|
ringinvval.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
ringinvval.p |
⊢ ∗ = ( .r ‘ 𝑅 ) |
3 |
|
ringinvval.o |
⊢ 1 = ( 1r ‘ 𝑅 ) |
4 |
|
ringinvval.n |
⊢ 𝑁 = ( invr ‘ 𝑅 ) |
5 |
|
ringinvval.u |
⊢ 𝑈 = ( Unit ‘ 𝑅 ) |
6 |
|
eqid |
⊢ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) = ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) |
7 |
5 6
|
unitgrpbas |
⊢ 𝑈 = ( Base ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) |
8 |
5
|
fvexi |
⊢ 𝑈 ∈ V |
9 |
|
eqid |
⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 ) |
10 |
9 2
|
mgpplusg |
⊢ ∗ = ( +g ‘ ( mulGrp ‘ 𝑅 ) ) |
11 |
6 10
|
ressplusg |
⊢ ( 𝑈 ∈ V → ∗ = ( +g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) |
12 |
8 11
|
ax-mp |
⊢ ∗ = ( +g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) |
13 |
|
eqid |
⊢ ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) |
14 |
5 6 4
|
invrfval |
⊢ 𝑁 = ( invg ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) |
15 |
7 12 13 14
|
grpinvval |
⊢ ( 𝑋 ∈ 𝑈 → ( 𝑁 ‘ 𝑋 ) = ( ℩ 𝑦 ∈ 𝑈 ( 𝑦 ∗ 𝑋 ) = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) ) |
16 |
15
|
adantl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ) → ( 𝑁 ‘ 𝑋 ) = ( ℩ 𝑦 ∈ 𝑈 ( 𝑦 ∗ 𝑋 ) = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) ) |
17 |
5 6 3
|
unitgrpid |
⊢ ( 𝑅 ∈ Ring → 1 = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) |
18 |
17
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑈 ) → 1 = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) |
19 |
18
|
eqeq2d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑈 ) → ( ( 𝑦 ∗ 𝑋 ) = 1 ↔ ( 𝑦 ∗ 𝑋 ) = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) ) |
20 |
19
|
riotabidva |
⊢ ( 𝑅 ∈ Ring → ( ℩ 𝑦 ∈ 𝑈 ( 𝑦 ∗ 𝑋 ) = 1 ) = ( ℩ 𝑦 ∈ 𝑈 ( 𝑦 ∗ 𝑋 ) = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) ) |
21 |
20
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ) → ( ℩ 𝑦 ∈ 𝑈 ( 𝑦 ∗ 𝑋 ) = 1 ) = ( ℩ 𝑦 ∈ 𝑈 ( 𝑦 ∗ 𝑋 ) = ( 0g ‘ ( ( mulGrp ‘ 𝑅 ) ↾s 𝑈 ) ) ) ) |
22 |
16 21
|
eqtr4d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ) → ( 𝑁 ‘ 𝑋 ) = ( ℩ 𝑦 ∈ 𝑈 ( 𝑦 ∗ 𝑋 ) = 1 ) ) |